Towards optimal Steiner tree routing in the presence of rectilinear obstacles

ABSTRACT

An apparatus and method for locating a good approximation of optimal Steiner tree routing in the presence of rectilinear obstacles, including finding a Steiner tree on an escape graph. The escape graph is constructed by forming lines from given points (pins) and obstacles. Obstacles and the segments of obstacles are provided with lines parallel to that segment at a given minimum distance S min  from the obstacle is constructed until it reaches either a boundary of an obstacle or a boundary of the core. For pins which do belong to a boundary of an obstacle, a ray, perpendicular to the segment of the boundary on which the pin is located is constructed from the pin and out from the obstacle until it reaches another obstacle or a boundary of the core. For pins which do not belong to an obstacle, vertical and horizontal lines are constructed. A Steiner tree may then be found on the escape graph by using any number of algorithms such as algorithm S and algorithm M. The solution to the problem of finding a Steiner tree for the escape graph also provides a suitable approximation of a Steiner tree for the original problem. This apparatus or method may be used to optimize the routing of conductive paths on integrated circuits.

This is a continuation application of U.S. patent Ser. No. 08/599,764, filed Feb. 12, 1996, now U.S. Pat. No. 5,615,128, which is a continuation of U.S. patent application Ser. No. 08/131,770, filed Oct. 4, 1993, now U.S. Pat. No. 5,491,641.

FIELD OF THE INVENTION

This invention relates to devices and methodologies for finding rectilinear Steiner trees in the presence of obstacles.

BACKGROUND OF THE INVENTION

Modern integrated circuits such a VLSI devices are typically made up of a large number of functional components formed on a single circuit substrate, or die. An increasingly important aspect of integrated circuit design concerns the placement of these functional components and the organization of interconnective paths that provide for the transmission of electrical signals between these functional components. Various algorithms have been developed to optimize the layout and interconnection of the functional components in an integrated circuit. The design of suitable interconnective pathways on an integrated circuit is typically termed routing. One goal of routing is to connect the functional components of the chip using the shortest possible conductive paths available.

A class of paths termed Steiner trees has been developed as one method that is used in the physical design of integrated circuits to efficiently route multi-terminal interconnective nets. A Steiner tree for n demand points is a tree (a connected graph with no closed paths) made up of lines that interconnect all n demand points of the tree. A Steiner tree, unlike for example a spanning tree, may also contain additional vertices that are not among the n demand points, in order to achieve a shorter pathway among these n demand points. A rectilinear Steiner tree (RST) of n demand points may be characterized as a tree composed only of orthogonal line segments (typically termed edges) that interconnect all n demand points (which are located at vertices). a rectilinear Steiner tree (RST) is confined to an underlying grid type graph which has traditionally been defined as the intersections of orthogonal lines (usually horizontally and vertically oriented) that are drawn through the n demand points. A graph may be considered a pair of sets G=(V,E), where V is set of vertices or points, and E is a set of edges between the vertices. Finding a minimum rectilinear Steiner tree (MRST) can be characterized as finding a Steiner tree whose edges are constrained to rectilinear shapes that in combination connect all of the desired points in the shortest path available.

Preferably, the interconnective paths of the various functional devices in an integrated circuit do not cross through the functional devices themselves. These functional components therefore represent obstacles on a chip that must be considered in routing. It is therefore advantageous to consider the general problem of finding an MRST in the presence of such obstacles. A rectilinear obstacle (referred to as "obstacle" in the remainder of this application) can be considered a region in a plane bounded by a polygon whose sides are parallel to some coordinate system. For example, all of the sides of the obstacle may be vertically or horizontally oriented (see FIG. 1). The problem of finding an MRST is a specific case of the more general problem of finding an MRST in the presence of rectilinear obstacles (MRSTO) with the number of obstacles equal to zero (0). Although there is extensive literature on the MRST problem, there previously was no solution for the general MRSTO problem.

BRIEF SUMMARY OF THE INVENTION

A principal object of the present invention is to provide a device and method for obtaining a good approximation of a minimum rectilinear Steiner tree in the presence of rectilinear obstacles. In accordance with the present invention, apparatus and methods for finding an optimal Steiner tree routing path in the presence of rectilinear obstacles includes finding a Steiner tree on a uniquely defined graph. This graph, termed herein an escape graph, is constructed by forming lines from the set of preselected demand points to be interconnected (hereinafter termed pins) and the obstacles present on the integrated circuit die. For each obstacle and for each segment of a boundary to an obstacle, a line is constructed parallel to that segment at some distance S_(min) (a predetermined non-negative real number termed herein the minimum spacing) from the obstacle. Currently, actual physically implemented VLSI circuits usually require some minimum allowable distance between the boundary of a functional component representing an obstacle and the nearest conductive track following the pathway determined by the routing process. Where such a minimum allowable distance is not required, the value of S_(min) may be zero.

The line being drawn parallel to the segment of the obstacle extends until it reaches either a boundary of an obstacle or a boundary of the region in which all the pins and obstacles reside, termed the core. For each pin that does not reside at the boundary of an obstacle, lines are drawn parallel to an appropriately oriented coordinate system, extending from the pin until they reach another obstacle or the edge of the core. For each pin located at a boundary of some obstacle, a ray (a line extending in only one direction) is drawn perpendicular to the boundary at which the pin resides, extending away from the obstacle until another obstacle or a boundary of the core is reached. Any algorithm suitable for determining a Steiner tree may then be applied to this escape graph to find an approximation of the minimum Steiner tree for the original problem, which included the obstacles.

Although the methods discussed herein focus on an optimizing process according to length and distance, any weights assigned to the edges of the graph may also be used for the purpose of eliminating or reducing congestion on the integrated circuit chip during the routing of a set of nets. Since the existence of at least some obstacles is almost always unavoidable in real integrated circuit designs, one advantage of the present invention is that it rapidly provides a good approximation of a MRSTO.

Other objects, features and advantages of the present invention will become apparent from a consideration of the following detailed description, and from the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a top view of a rectilinear obstacle;

FIG. 2 is a top view of a core with rectilinear obstacles and pins from which an escape graph and a rectilinear Steiner tree has been constructed;

FIG. 3A is a bounding box constructed around the given pins for algorithm M;

FIG. 3B is a net routed by a maze expansion algorithm;

FIG. 4 is a flowchart of algorithm M;

FIG. 5 is a graph divided into Voronoi regions used in algorithm S.

FIG. 6 is a flowchart of a method of creating an escape graph (G) within a core using a single microprocessor;

FIG. 7 is a flowchart of a method for creating an escape graph (G) within a core using by multiple microprocessors; and

FIG. 8 is a flowchart of algorithm S.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

An apparatus and method of the present invention for finding a good approximation of an optimal Steiner tree routing in the presence of rectilinear obstacles includes finding a Steiner tree in a uniquely constructed graphs termed an escape graph. FIGS. 6 and 7 show the details of constructing an escape graph (physically shown in FIG. 2) by forming escape lines (or conductive paths) from given points (pins) and obstacles. Any algorithm suitable for determining Steiner trees on a graph is then applied to the escape graph to find an approximation of a minimal Steiner tree for the original problem. Examples of two such algorithms are detailed in FIGS. 4 and 8 and described more fully below.

Referring to, FIG. 1 there is shown a depiction of a core 19 with three rectilinear obstacles 10, 12, and 14. These rectilinear obstacles are regions in the plane bounded by a polygon whose sides are parallel to some Cartesian coordinate system. These obstacles may be rectangular 10 or have one or more protrusions as illustrated at fingers 16 or one or more inlets as illustrated at 18.

To construct an escape graph, such as the graph shown in FIG. 2, a Cartesian coordinate system may be established in a plane so that all coordinates may be given with respect to this coordinate system. The distance d (P₁, P₂) between any two points P₁ and P₂ in this escape graph may be considered the rectilinear or Manhattan distance between the points.

As shown in FIG. 2, the core is defined by boundaries 62, 64, 66, and 68. Within these boundaries are predetermined rectilinear obstacles, such as obstacles 20, 22, and 24 having vertical and horizontal sides. A predetermined set of pins 30, 32, 34, 36, and 38 are also shown in FIG. 2.

From the predetermined set P of pins (such as pins 30, 32, 34, 36 and 38) and the predetermined set O of obstacles (such as obstacles 20, 22 and 24), a grid system of escape lines may be constructed to form the escape graph. These escape lines may be termed either p-escape lines, induced by the pins, or termed o-escape lines, induced by the obstacles.

One or more processing means may be used to determine whether there are obstacles 200, pins on the boundary of an obstacles 210, or pins that are not on the boundary of an obstacle 212. Orthogonal escape lines are constructed accordingly (as discussed more fully below). FIGS. 6 and 7 illustrate flowcharts for the operations to be performed by the one or more processors. FIGS. 6 and 7 are meant to be exemplary, and are not meant to limit the number of processors which may be used, the order in which the determination of the existence of remaining obstacles or pins is accomplished, or the order in which the escape lines or conductive paths are formed.

More specifically, the escape graph is constructed as follows. For each obstacle and each segment of the boundary of an obstacle, a line parallel to that segment is constructed at a distance S_(min) (a given non-negative real number) from the obstacle. This line extends until it reaches either a boundary of an obstacle or a boundary of the core. The value of S_(min) reflects the minimum allowable spacing, for example, between a functional components in an integrated circuit and an adjacent conductive pathway. In appropriate instances the value of S_(min) may be zero. This set of lines parallel to the boundaries of the obstacles defines the set of o-escape lines. A set of exemplary escape lines is shown in FIG. 2. At the obstacle 22, for example, a segment of the polygon representing the obstacle 22 is selected, such as the top segment 22a, and an escape line 23 is constructed parallel to the horizontal top segment 22a. This escape line 23 extends in both directions to the boundaries 64 and 68 of the core since there is no obstacle to block the escape line. Each segment of the obstacle is considered and escape lines are drawn parallel to the segments of the obstacle. As noted above these o-escape lines spaced some non-zero distances S_(min) from the segment of the obstacle.

Along with the obstacles, the predetermined pins p residing on the boundaries of obstacles are also considered. Through each such pin, a ray is constructed perpendicular to the segment of the boundary on which the pin resides, extending from the pin and out from the obstacle until it reaches another obstacle or a boundary of the core. This arrangement of escape lines is also shown in FIG. 2. Pin 32, for example, is located on a vertical segment of obstacle 22. An escape line in constructed perpendicular to that segment until it reaches another segment of obstacle 22. Another example is pin 34 located on a vertical segment of obstacle 24. As shown, an escape line is similarly constructed perpendicular to that segment until it reaches the boundary 64 of the core.

As noted above, the pins p that do not reside on any obstacle are also considered. Through each such pin a pair of mutually orthogonal lines are drawn, each line extending to either an obstacle or a boundary of the core. Referring again to FIG. 2, orthogonal escape lines can be seen to extend from pin 36. Thus a vertical escape line 37 extends from pin 36 to the lower boundary 62 and upper boundary 66 of the core and an escape line 39 extends horizontally from pin 36 to obstacle 24 and boundary 68 and the set of escape lines drawn from pins that are located either on the boundary of an obstacle or standing freely are termed the set of p-escape lines.

The escape graph G formed from the p-escape lines and the o-escape lines may also be considered as a set of vertices V and a set of line segments of edges E. The escape graph can thus be considered equivalent to a set of the intersection points of escape lines V (for example, 50, 52, 54, and 56 as shown in FIG. 2) and the segments of the escape lines E connecting any two of these intersection points (for example, 58 and 60 as shown in FIG. 2). This statement may be represented mathematically as G=(V,E).

The inventors have determined that for the type of MRSTO problem discussed above, an MRSTO tree is embedded within the escape graph developed by the above methodology. Although any suitable routing algorithm may be used to find a minimum Steiner tree in the escape graph developed above, two exemplary algorithms are disclosed herein as example of suitable algorithms for finding an approximation of a minimum Steiner tree in the escape graph. The first algorithm is a maze-based algorithm, "algorithm M," and the second is a Steiner tree based algorithm that starts from a uniquely configured spanning tree, "algorithm S." Experimental results have shown that processor based performance of algorithm S runs faster than performance of algorithm M for nets having more than 7 pins (137 times faster for a net with 1694 pins) while the total path length is at most two percent (2%) longer than the path length produced by algorithm M (in some cases algorithm S even produced a shorter path length than algorithm M). As discussed above the present invention is deemed to include application of either of the M or S algorithms, as well as other algorithms that are known to be used, or may be used to find an approximation of a minimum Steiner tree in the escape graph.

Algorithm M

Algorithm M is a maze router type algorithm that operates in a graph instead of a plane. The steps of Algorithm M are briefly described below and illustrated in FIG. 4. An input 130 to algorithm M includes the pins P and the escape graph found by the method discussed above and illustrated in FIGS. 6 and 7. In operation, a bounding box is first constructed around the set of pins P as illustrated at 132 in FIG. 4. An exemplary bounding box 82 is shown is FIG. 3A. The next step 134 involves selection of the pin that is the closest to the center of the bounding box surrounding all of the pins. This pin becomes the center point of a Steiner tree T that is being constructed. The center point 80 or first point on the Steiner tree T is shown in FIG. 3A.

A maze expansion step 138 is performed from the existing Steiner tree T (initially a single point 80), until another pin p that is not in T is reached. Any maze routing algorithm may be used to accomplish the maze expansion, such as for example, Lee's algorithm, Soukup's algorithm, and Hadlock's algorithm. These algorithms are meant as examples only and are not meant to limit the invention in any way. In Lee's algorithm, the central or source pin such as pin 80 in FIG. 3 is labeled as vertex 0. Each vertex in the escape graph connected by a single edge to vertex 0 is then labeled as some minimum weighting, such as 1. Vertices adjacent to these labeled 1 vertices are then labeled with a greater weighting value, such a 2. This labeling process continues until another pin in the boundary box has been reached. An example of this methodology is shown in FIG. 3B. Pins are connected starting at a pin with vertex having a value of N, and proceeding to another vertex with value of N-1 until the vertex of value 0 has been reached. One exemplary path is shown in FIG. 3B. Other paths could, of course, be found. Once two of the pins have been connected, such as the pin 51 at X₀ and pin 53 at X₃ in FIG. 3b, the values of the remaining vertices are reassigned. Each vertex along the Steiner tree path connecting X₀ and X₃ is reassigned the value of 0. Each vertex immediately adjacent the pins X₀ and X₃, as well as each vertex located adjacent to the Steiner tree path connecting X₀ and X₃ is now assigned a value of 1. Vertices further from the Steiner tree are similarly assigned new values. Once new values assigned to these vertices have been reassigned, another pin can be connected to the Steiner tree connecting the pins at X₀ and X₃. The additional path connecting this next pin with X₀ and X₃ is again determined by moving from a vertex having a value N, to a vertex having a value N-1, and then to a vertex having a value N-2, until the path is established. The procedure of recalculating the values to be assigned to each of the vertices is repeated, and a path to the next pin is determined, until all of the pins are connected.

A flow chart for processing means implementing the methodology of the present invention including the steps to be performed in algorithm M is illustrated in FIG. 6. After constructing the escape graph in the manner discussed above, the method illustrated in the flow chart of FIG. 6 further includes the steps of determining 140 a path from tree T to a pin p and including this path in the Steiner tree T searching 136 to find if any remaining pins that are not yet connected to the tree. When there are no pins remaining, the Steiner tree has been found as shown at decision step 142. If there are unconnected pins remaining, the maze expansion 138 process described above is repeated. This process continues until all pins have been connected with the Steiner tree T.

Since the size of the escape graph is usually much larger than the number of pins, the complexity of this algorithm is typically considered to be O (n(N+K)²). The resulting Steiner tree obtained by implementing algorithm M is usually close to optimal, since the algorithm restarts the maze expansion whenever it includes a pin in the Steiner tree.

Algorithm S

Algorithm S is a Steiner tree based algorithm that employs the development of a spanning tree as a step in the formation of a minimum Steiner tree approximation. The steps of Algorithm S are briefly described below and a flow chart for implementing algorithm S is illustrated in FIG. 8.

As noted above, an escape graph is first determined from the predetermined locations of the pins and the obstacles in the core. The locations of electrical interconnection points on an integrated circuit die and the locations of functional components on the integrated circuit could, for example, provide the locations of the pins and obstacles in the core. Application of algorithm S to the escape graph discussed above also provides a methodology for determining an approximation of a minimum Steiner tree that interconnects all of the predetermined pins. Since the escape graph is constructed with the presence of obstacles, the approximation of a minimum Steiner tree developed on the escape graph provides a minimum rectilinear Steiner tree interconnecting the pins in the core even in the presence of the obstacles.

Application of algorithm S to the escape graph constructed in the manner discussed above initially requires subdividing the escape graph into regions about each pin. These regions are termed Voronoi regions. The Voronoi regions divide up the vertices of the escape graph so that all of the vertices closest to any particular pin are assigned to the Voronoi region associated with that pin. Referring to FIG. 5, there is shown an escape graph 151 divided into three exemplary Voronoi regions, including a Voronoi region N(q) at 160 that is associated with pin q at 161, Voronoi region N (r) at 162 associated with pin r at 152, and Voronoi region N (p) at 164 associated with pin p at 154. As shown in FIG. 5, each of vertex t at 155, vertex u at 156, and vertex s at 157 reside in Voronoi region N (p) at 164 since each of vertices s, t and v reside closer to pin p at 154 then any of the other pins in the escape graph, such as pin r at 152 and pin q at 161. If a vertex is at an equal distance to two separate pins, this vertex is arbitrarily assigned to one Voronoi region or another.

Application of algorithm S further requires formation of an additional graph termed graph G₁. This graph contains a set of vertices corresponding to all of the demand points or pins from the set P of the escape graph G that are to be interconnected. The graph G₁ also includes a set of edges E₁ that directly connect some, but not necessarily all, of the pins P of the graph G₁. The set of edges E₁ in the additional graph G₁ are constructed by reference to the escape graph G and the Voronoi regions constructed on the escape graph. Any two pins p and q in the graph G₁ are to be directly connected by an edge in E₁ if the corresponding pins p and q in the escape graph G have a path between them that includes an edge bridging the boundary between the two Voronoi regions in the escape graph G around the pins p and q in G. That is, one of the vertices v and u forming the end points of the edge bridging the boundary between the Voronoi regions N(p) and N(q) must reside in one of these Voronoi regions and the other vertex must reside in the other Voronoi region. Referring again to FIG. 5, edge 159 having an end point u at vertex 156 and an end point s at vertex 157 would not be selected since both of the end points v and s of edge 159 reside at vertices 156 and 157 within a single Voronoi region N(p) at 164. Edge 163, however, includes an end point v at 158 located in Voronoi region N(q) at 160 and an end point us at vertex 156 located in Voronoi region N(p) at 164.

Once the set of edges E₁ has been constructed, there is associated with each of these edges in E₁ a number or weighting factor. This weighting factor may, but need not necessarily, be associated with a minimum path distance in G between the two pins in G corresponding to the two pins in G₁ that are directly connected. This minimum path must also include the edge in G bridging the boundaries of the Voronoi regions of the pins in G. Mathematically this minimum path distance can be represented by the equation:

    d'.sub.1 (p,q)=min{d.sub.1 (p,u)+d(u,v)+d.sub.1 (v,q)}.

Where d₁ (p,u) is a minimum path from vertices p to u, and d₁ (v,q) is a minimum path from vertices v to q. The term d(u,v) is a weight or distance associated with the edge in the escape graph G having the end points u and v. Typically a shorter path may be found between any two pins, such as pins p and q, but this shorter path may not necessarily include the edge bridging the boundaries of the relevant Voronoi regions.

After the set of edges E₁ in the graph G₁ has been constructed and weights have been assigned to these edges, a minimum spanning T₁ is constructed from the graph G₁, based on the weights associated with the edges of the set E₁. Several minimum spanning trees may be found in G₁. In this instance any one of these minimum spanning trees may be arbitrarily selected.

Once the minimum spanning tree T₁ has been constructed, a new subgraph G_(s) is constructed from the escape graph. This subgraph G_(s) is constructed by selecting the shortest paths in G connecting the pins in P corresponding (for example, corresponding by weight) to the edges from the minimum spanning tree T₁ in the graph G₁. If there is more than one minimum path in G between any two connected pins that also satisfies this condition, then any one of these minimum paths in G may be arbitrarily selected. A minimum spanning tree T_(s) is then constructed from the subgraph G_(s). If more than one minimum spanning tree T_(s) exists, then again one may be arbitrarily selected. After the minimum spanning tree T_(s) has been constructed, a Steiner tree is constructed from this minimum spanning tree T_(s). This Steiner tree is constructed by deleting from the minimum spanning tree T_(s) end points of the tree T_(s) (sometimes termed leaves) residing at vertices that are not demand pins. The edges connecting the deleted end points or leaves are also removed.

Using algorithm S for an MRSTO problem, a RSTO can be determined whose total length of all edges is at most 2(1-1/l)times that of a MRSTO, where l is the number of leaves in the MRSTO. The total complexity of this approach is O((n+K)² log(n+k)) times. Using algorithm S thus provides a faster method of finding a minimum Steiner tree in the escape graph than algorithm M when the number of pins n is sufficiently large.

In conclusion, it is to be understood that the present invention is not to be limited to that precisely as described hereinabove and as shown in the accompanying drawings. More specifically, any method of finding a Steiner tree in a graph may be used, as the algorithms specifically addressed are examples of these methods. Further, any weights, including but not limited to distance, for edges of the graph may be used for the purpose of eliminating or reducing the congestion on the chip during the routing of a set of nets. Accordingly, the present invention is not limited to the arrangements precisely as shown and described hereinabove. 

What is claimed is:
 1. A method of forming a graph for use in routing interconnections on an integrated circuit layout, wherein said integrated circuit layout is bounded by a core and includes obstacles and pins, said method comprising:forming a first set of lines parallel and in proximity to corresponding segments of boundaries of said obstacles, wherein the corresponding segments of the boundaries include at least one segment that is not an extreme top, extreme bottom, extreme right or extreme left edge of its corresponding obstacle, and wherein the lines of said first set of lines extend until either a boundary of one of said obstacles or a boundary of said core is reached; and forming a second set of lines that extend from pins residing on the boundaries of said obstacles in a direction perpendicular to and out from respective boundaries of said obstacles until either a boundary of one of said obstacles or a boundary of said core is reached.
 2. A method according to claim 1, wherein the first set of lines includes a line parallel and in proximity to each segment of the boundaries of said obstacles.
 3. A method according to claim 1, wherein the second set of lines includes a line that extends from each pin residing on the boundary of each of said obstacles.
 4. The method of claim 1, further comprising:forming a third set of mutually orthogonal lines through pins that do not reside on any of said obstacles and extending until either a boundary of one of said obstacles or a boundary of said core is reached.
 5. A method according to claim 4, wherein two mutually orthogonal lines are formed through each pin that does not reside on any of said obstacles.
 6. The method of claim 4, wherein said obstacles are rectilinear obstacles.
 7. The method of claim 6, wherein said segments of the boundaries of said rectilinear obstacles are parallel to a Cartesian coordinate system.
 8. The method of claim 7, wherein the segments of the core boundary are parallel to a Cartesian coordinate system.
 9. The method of claim 4, wherein the step of forming said first set of lines includes the step of forming said first set of lines in a manner that the distance between said first set of lines and corresponding segments of said obstacles is zero.
 10. The method of claim 4, wherein the step of forming said first set of lines includes the step of forming said first set of lines in a manner that the distance between said first set of lines and corresponding segments of said obstacles is given by a minimum allowable spacing.
 11. A method of routing interconnects for an integrated circuit layout that is bounded by a core includes obstacles and pins, said method comprising:forming an escape graph that includes the steps of:forming a first set of lines parallel and in proximity to corresponding segments of the boundaries of said obstacles, wherein the lines of said first set of lines extend until either a boundary of one of said obstacles or a boundary of said core is reached, and forming a second set of lines that extend from pins residing on the boundaries of said obstacles in a direction perpendicular to and out from respective boundaries of said obstacles until either a boundary of one of said obstacles or a boundary of said core is reached; and finding a Steiner tree in said escape graph.
 12. The method of claim 11, wherein the step of forming said escape graph further includes the step of forming a third set of mutually orthogonal lines through pins that do not reside on any of said obstacles and extending until either a boundary of one of said obstacles or a boundary of said core is reached.
 13. The method of claim 12, wherein said obstacles are rectilinear obstacles.
 14. The method of claim 13, wherein said segments of the boundaries of said rectilinear obstacles are parallel to a Cartesian coordinate system.
 15. The method of claim 14, wherein the segments of the core boundary are parallel to a Cartesian coordinate system.
 16. The method of claim 12, wherein the step of forming said first set of lines includes the step of forming said first set of lines in a manner that the distance between said first set of lines and corresponding segments of said obstacles is zero.
 17. The method of claim 12, wherein the step of forming said first set of lines includes the step of forming said first set of lines in a manner that the distance between said first set of lines and corresponding segments of said obstacles is given by a minimum allowable spacing.
 18. The method of either of claims 11 or 12, wherein said step of finding a Steiner tree includes the step of finding a Steiner tree using a spanning tree algorithm ("algorithm S").
 19. The method of claim 18, wherein said spanning tree algorithm includes the steps of:(a) constructing Voronoi regions in said escape graph; (b) constructing a first graph including said pins, and edges connecting two of said pins respectively in said escape graph which extend between adjacent Voronoi regions; (c) constructing a second graph as a minimum spanning tree of said first graph; (d) constructing a third graph as a subgraph of said second graph including shortest paths in said escape graph corresponding to edges in said second graph; (e) constructing a fourth graph as a minimum spanning tree of said third graph; and (f) constructing said approximate Steiner tree from said fourth graph by deleting said point which constitute end points in said fourth graph and do not correspond to said pins.
 20. The method of claim 19, wherein:step (b) further comprises computing weights for said edges in accordance with lengths thereof in said escape graph; and step (c) further comprises constructing said second graph in accordance with said weights.
 21. The method of either of claims 11 or 12, wherein said step of finding of Steiner tree includes the step of finding a Steiner tree using an maze-based algorithm ("algorithm M").
 22. The method of claim 21, wherein said maze-based algorithm includes the steps of:(a) constructing a bounding box enclosing said pins; (b) selecting a source pin among said pins in a manner that said source pin is closest to the center of said bounding box, said source pin forming said initial point of said Steiner tree; (c) performing a maze expansion from said Steiner tree to include another pin that is not in said Steiner tree; and (d) performing step c until all of said pins in said circuit layout is included in said Steiner tree.
 23. The method of claim 22, wherein said step of performing said maze expansion includes the step of performing a maze expansion in accordance with Lee's algorithm.
 24. The method of claim 22, wherein said step of performing said maze expansion includes the step of performing a maze expansion in accordance with Soukup's algorithm.
 25. The method of claim 22, wherein said step of performing a maze expansion includes the step of performing a maze expansion in accordance with Hadlock's algorithm. 